Maximal element of a finite predicate above a witness
ProvedWilliamsonShmoys.finite_predicate_maximal_extension__p2m_6802ab23_a3ac_43c6_8aa9_163dea1542b1_a1approximation-algorithmsplanar-graphstreewidth
In a finite partial order, any element satisfying a predicate lies below a predicate-satisfying element that is maximal among predicate-satisfying elements above it.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_predicate_maximal_extension__p2m_6802ab23_a3ac_43c6_8aa9_163dea1542b1_a1
{α : Type} [Fintype α] [PartialOrder α]
(P : α → Prop) (a : α) (ha : P a) :
∃ b, a ≤ b ∧ P b ∧ ∀ c, b ≤ c → P c → c = b := by sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).